Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and

Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and

difference system with delay

Abdennasser Chekroun

Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, University of Tlemcen, Tlemcen 13000, Algeria

Received 27 April 2019, appeared 19 November 2019 Communicated by Tibor Krisztin

Abstract. In this paper, we consider a general study of a recent proposed hematopoietic stem cells model. This model is a combination of nonlocal diffusion equation and difference equation with delay. We deal with the properties of traveling waves for this system such as the existence and asymptotic behavior. By using the Schauder’s fixed point theorem combined with the method based on the construction of upper and lower solutions, we obtain the existence of traveling wave fronts for a speedc>c^?. The case c=c^? is studied by using a limit argument. We prove also thatc^? is the critical value.

We finally prove that the nonlocality increases the minimal wave speed.

Keywords:traveling wave front, nonlocal diffusion systems with delay, difference equa- tion, monostable equation.

2010 Mathematics Subject Classification: 35C07, 35K08, 35K57.

1 Introduction

Propagation and invasion phenomena are often analyzed through the study of traveling wave.

A traveling wave is a solution of special form and it can be seen as an invariant function with respect to spatial translation describing processes. Many researchers have used such solutions to model the dynamics of biological invasions and the spread of population (see [25,31] and references therein). The theory of these solutions has been widely developed for the reaction- diffusion equations and there has been some success studied for establishing the existence of traveling wave for the reaction-diffusion equations with or without delayed local or nonlocal nonlinearity (see [1,5,7,11,13,15,17,23–26,30,34,35] and references therein). On the other hand, there have been studies about traveling waves for nonlocal diffusion systems where the diffusion is described by integral, see [6,8–10,20,21,32,33,36].

Recently in [1], a new model (based on the model of Mackey [18]) describing hematopoiesis was presented and discussed. This model is the following coupled reaction-diffusion and

BEmail: abdennasser.chekroun@mail.univ-tlemcen.dz

difference system (see also [16] for a particular case)



















∂N(t,x)

∂t =D∂²N(t,x)

∂x² −(δ(N(t,x)) +β(N(t,x)))N(t,x) +2(1−K)e^−γr

Z +_∞

−_∞ Γ(r,x−y)u(t−r,y)dy, u(t,x) = β(N(t,x))N(t,x) +2Ke^−γr

Z +_∞

−_∞ Γ(r,x−y)u(t−r,y)dy.

(1.1)

This system (1.1) describes mature-immature blood cells interaction. The blood cell population is split into two compartments of mature and immature cells. Each compartment represents the cells in resting phase and proliferating phase, respectively. In this system,Nrepresents the density of resting cells anduthe density of new active or proliferating cells (see also [2–4,18]).

As mentioned, a special case of the above system is treated in [16] which corresponds to K = 0. The one-dimensional domain was taken. The positive coefficients D andd represent the diffusion rates in the quiescent and proliferating phase, respectively. The delay r > 0 describes the duration of the active phase andγ > 0 a programed cell death rate. The terms 2(1−K)e^−γr and 2Ke^−γr, for 0 ≤ K < 1, describe the part of divided cells (coefficient 2 represents the division) that enter the quiescent and proliferating phase, respectively. The nonlinearities are given by(δ(x) +β(x))xand β(x)x whereδ is a natural death rate and βis the rate of flux between the both phases (see [1] for more interpretations of parameters).

The system (1.1) shows the non-local effect that is caused by cells diffusing (with a rated) during proliferating phase whereΓdenote the Green’s function given by

Γ(t,x) = ¹ 2√

dπtexp

− ^x

2

4dt

, t>0, x ∈_R.

We note thatΓsatisfies

Z +_∞

−_∞ Γ(t,x)dx=1, t>0. (1.2) In the case where K = 0, the system (1.1) is equivalent to the following one dimensional scalar delayed reaction-diffusion equation

∂N

∂t (t,x) =D∂²N(t,x)

∂x² −(δ(N(t,x)) +β(N(t,x)))N(t,x) +2e^−γr

Z +_∞

−_∞ Γ(r,x−y)β(N(t−r,y))N(t−r,y)dy.

(1.3)

When d 7→ 0⁺ and using the heat kernel property, the nonlocal term in (1.3) becomes the following expression

β(N(t−r,x))N(t−r,x) = lim

d7→0⁺

Z +_∞

−_∞ Γ(r,x−y)β(N(t−r,y))N(t−r,y)dy.

This case corresponds to a problem with a local nonlinearity. Moreover, if r = 0 then the system (1.3) is reduced to

∂N

∂t (t,x) =D∂²N(t,x)

∂x² −δ(N(t,x))N(t,x) +β(N(t,x))N(t,x).

Consideringδ(x) =xandβ(x) =1, we get the classical Fisher–KPP equation [12,14,22,25].

In [1], the authors considered a hematopoietic dynamics model that took into account spatial diffusion of cells where the Laplacian operator∆:=∂²/∂x²is local. This suggests that the influence is caused by the neighborhood variations. As in many areas, when the density of the considered population is not small, such as the dynamics of cells, the local diffusion is not sufficiently accurate (see, [19]). Moreover, it is emphasized that the nonlocal operator has some properties of the Laplacian one and is reduced in some cases to it (see, [21]). In this work, we deal with the case of nonlocal diffusion which means that we replace the local Laplacian operator by the following convolution nonlocal diffusion

(h∗v)(t,x)−v(t,x):=

Z +_∞

−_∞ h(x−y)[v(t,y)−v(t,x)]dy, with h:R→_Ris a nonnegative function satisfying

h(x) =h(−x) forx∈_R,

Z +_∞

−_∞ h(x)dx=1,

and Z _+∞

−_∞ h(x)e^−λxdx<+_∞, for any λ>0.

We shall focus on the following coupled nonlocal diffusion and difference system with delay, fort >_{0 andx} ∈_R,



















∂N(t,x)

∂t = D[(h∗N)(t,x)−N(t,x)]−(δ(N(t,x)) +β(N(t,x)))N(t,x) +2(1−K)e^−γr

Z _+∞

−_∞ Γ(r,x−y)u(t−r,y)dy, u(t,x) =β(N(t,x))N(t,x) +2Ke^−γr

Z _+∞

−_∞ Γ(r,x−y)u(t−r,y)dy.

(1.4)

Our purpose is to prove the existence of fronts of the above system.

This paper is organized as follows. In the next section, we start by some preliminaries about the solution of the system. Section3is devoted to the proof of the existence of traveling wave fronts when the speed is greater or equal to a threshold denoted c^?. We also prove the nonexistence when the speed is less than c^?. We proceed by giving a result about the monotonicity of the critical speed wave with respect to the diffusion parameters. Section 4is devoted to the discussion.

2 Preliminaries

Let X= BUC(_R,R)be the Banach space of all bounded and uniformly continuous functions fromRtoRwith the usual supremum norm|·|_XandX⁺:={φ∈ X: φ(x)≥0, for allx∈_R}. The space Xis a Banach lattice under the partial ordering induced by the closed coneX⁺.

We set

(T(t)ω)(x):=

Z

RΓ(t,x−y)ω(y)dy, t>0, x ∈_R.

Then, we get (see for instance [29]) an analytic semigroupT(t): X → Xsuch that T(t)X⁺ ⊂ X⁺, for allt≥0.

Throughout this paper, we make the following hypotheses on the functionsβandδ.

The functionN7→ β(N)is continuously differentiable onRand decreasing onR⁺

with lim_N→+_∞β(N) =0. (2.1)

The functionN7→δ(N)is continuously differentiable onRand increasing onR⁺. (2.2) In [1], the authors studied mainly the existence of traveling wave fronts by using the monotone iteration technique coupled with the sub- and super-solutions method developed in [30]. For the system (1.4), we shall use the same technique based on the construction of upper and lower solutions. Recall that a traveling wave of (1.4) is a solution of special form

(N(t,x)_,u(t,x)) = (φ(x+ct)_,ψ(x+ct))_,

whereφ,ψ∈C¹(_R,R⁺)andc>0 is a constant corresponding to the wave speed (see [19,25, 29]). We consider the functions f, g:R⁺ →_R⁺, defined by

f(_s) = (δ(_s) +β(_s))_{s and g}(_s) =β(_s)_{s, s}∈ _R⁺.

We setz= x+ctand substitute (N(t)(x),u(t)(x))with(φ(z),ψ(z))into (1.4). We obtain the corresponding wave system







cφ⁰(z) =D[(h∗φ)(z)−φ(z)]− f(φ(z)) +2(1−K)e^−γr(T(r)ψ) (z−cr),

ψ(z) = g(φ(z)) +2Ke^−γr(T(r)ψ) (z−cr). (2.3) The following proposition ensures existence of positive constant steady state under additional conditions. Recall that 0≤K <1, we suppose the following necessary condition,

2Ke^−γr<1. (2.4)

Proposition 2.1. Assume thatδ(0)>0. If

δ(0) + (1−2e^−γr)β(0)

2e^−γrδ(0) <K< ¹

2e^−γr, (2.5)

then(1.4)has two distinct steady states: (0, 0)and(N^?,u^?). If (2.5) does not hold, then(0, 0)is the only equilibrium of (1.4).

It is shown in [1] that there existsc^? >0 such that the system (2.3), with local diffusion or (1.1), has a monotone solution(φ,ψ)defined on R, for each c ≥ c^?, subject to the following asymptotic boundary condition

φ(−_∞) =ψ(−_∞) =0, φ(+_∞) = N^? and ψ(+_∞) =u^?, (2.6) where(N^?,u^?)is the only constant positive equilibrium of (1.1). In this case, the correspond- ing solution (N(t,x),u(t,x)) = (φ(x+ct),ψ(x+ct)) is called a traveling wave front with wave speedc>0 of (1.1). Such result needs the following assumptions.

The function N7→ g(N):= β(N)Nis increasing on[0,N^?], (2.7) β(N) +δ(N)≥ β(0) +δ(0), for all N∈[0,N^?]. (2.8) The main result of this paper is given in the following theorem where we show, under the same conditions as in [1], that there exists a minimal wave speed (of course other than that in [1]) for the existence of fronts for (1.4) (system with nonlocal diffusion).

Theorem 2.2. Assume that(2.1),(2.2),(2.4),(2.5),(2.7)and(2.8)hold. Then, there exists c^? >0such that for every c≥c^?,(1.4)has a traveling wave front which connects(0, 0)to the positive equilibrium (N^?,u^?). Let c∈(_0,c^?). Then, there is no traveling front of (1.4).

3 Existence of traveling wave fronts

In this section, we study the existence of traveling wave solutions of system (1.4). This is treated mainly by the Schauder’s fixed point theorem with the notion of upper and lower solutions and Laplace transform. Let A: X→Xbe the bounded linear operator defined by

(Aψ) (z) =2Ke^−γr(T(r)ψ) (z−cr), z∈ _R. (3.1) A direct computation leads to |A|_L(X) = 2Ke^−γr < 1. Then, the operator A is a contrac- tion. Thereby, ψcan be calculated explicitly according toφby considering the inverse of the operator Id−A. In fact, if we put, forz∈_R, k ∈_N,

ξ_k(z) = (2Ke^−γr)^k 2(kdπr)^{1/2 exp}

−(z−kcr)² 4kdr

, (3.2)

we have, for ϕ∈ X,

(Id−A)⁻¹(ϕ) =

+_∞ k

∑

A^kϕ= ξ∗ϕ, where

ξ(z) =

+_∞ k

∑

(2Ke^−γr)^k_Γ(kr,z−kcr) =

+_∞ k

∑

ξ_k(z), z∈_R. (3.3) The functionξ_k,k∈_Nsatisfies

Z +_∞

−_∞ ξ_k(y)dy= (2Ke^−γr)^k. The system (2.3) becomes an uncoupled system

(cφ⁰(z) =D[(h∗φ)(z)−φ(z)]− f(φ(z)) +₂(₁−K)e^−γr(T(r)ψ) (z−cr)_,

ψ(z) = (ξ∗g(φ))(z), (3.4)

with ξ given by (3.3). We can then write (3.4) as a single differential equation

cφ⁰(z) = D[(h∗φ)(z)−φ(z)]− f(φ(z)) +2(1−K)e^−γr[T(r) (ξ∗g(φ))] (z−cr). (3.5) It is clear that if(φ,ψ)is a monotonic solution of (2.3)–(2.6), thenφis a monotone solution of (3.5) and

φ(−_∞) =0 and φ(+_∞) = N^?. (3.6)

Under (2.4) and (2.7) we prove easily that even if φ is a monotone solution of (3.5)-(3.6), then (φ,ξ∗g(φ)) is a monotone solution of (2.3)–(2.6). Hence, we only need to consider the solutions of (3.5) subject to boundary condition (3.6).

Our objective is to show the existence of traveling wave front solutions for the coupled nonlocal diffusion and difference system (1.4). To this end, we use the method based on the notion of an upper and a lower solutions combined with Schauder’s fixed point theorem [21,27].

Let

C_[0,N^?_](_R,R) ={φ∈C(_R,R): 0≤φ(z)≤ N^?, z∈_R}. Define the operator H:C_[0,N^?_](_R,R)→C(_R,R)by

H(φ)(z) =D(h∗φ)(z) + (µ−D)φ(z)− f(φ(z)) +2(1−K)e^−γr[T(r) (ξ∗g(φ))] (z−cr), where µ > D+max_s∈[0,N^?_] f⁰(s) is a constant. Next, we show that H satisfies the condition given in the following lemma.

Lemma 3.1. Assume that (2.1), (2.2), (2.4), (2.5) and (2.7) hold. Then, H satisfies the following property

H(φ₁)(z)−H(φ₂)(z)≥0, for allφ₁,φ₂ ∈X⁺such that0≤ φ₂(z)≤φ₁(z)≤ N^?, for all z∈_R.

The proof of this lemma is easy to establish, so we omit the details here.

Now, it is easy to remark that (3.5) is equivalent to the following simplified equation, cφ⁰(z) =−µφ(z) +H(φ)(z). (3.7) Define the operatorF:C_[0,N^?_](_R,R)→C(_R,R)by

F(φ)(z) = ¹ c

Z _z

−∞e^{−µc(z−s)}H(φ)(s)ds.

We can easily see that the operator F is well defined satisfying (3.7) and the existence of solutions for (3.5) is changed into investigating the existence of a fixed point of operator F.

The following remark is a key to show the existence of such fixed point.

Remark 3.2. Lemma3.1 implies that either Hand also F are monotone for φ. Moreover, we can deduce that H(φ)(z)and F(φ)(z)are both nondecreasing inz ∈ _Rwith the assumption thatφ∈C_[0,N^?_](_R,R)is nondecreasing inz∈_R.

For 0<ν < ^µ_c, define

B_ν(_R,R) =ⁿφ∈C(_R,R): sup_z∈R|φ(z)|e^−ν|z| <+_∞^o. and the exponential decay norm

|φ|_ν =sup

z∈_R

|φ(z)|e^−ν|z|, for φ∈ Bν(_R,R). It is easy to check that(Bν(_R,R),| · |_ν)is a Banach space.

Now, we define the meaning of an upper and lower solutions of (3.5).

Definition 3.3. A continuous functionφ∈ C_[0,N^∗_](_R,R)is called an upper solution of (3.5) if φ⁰ exists almost everywhere (a.e.) and satisfy

cφ⁰(z)≥D[(h∗φ)(z)−φ(z)]− f(φ(z)) +2(1−K)e^−γr

T(r) ξ∗g(φ)(z−cr), a.e. inR.

A lower solution φ of (3.5) is defined in a similar way but it satisfies the above differential inequality in reversed order.

Existence of traveling wave front solutions need to find suitable upper φ and lower φ solutions of (3.5). For this purpose, we consider the transcendental characteristic function for the linearized problem of (3.5) near the zero solution. Let

λ⁺(c) = ^c 2d 1+

s 1+ ^4d

rc²ln e^γr

2K !

. Forλ∈(0,λ⁺(c)), we have

1−2Ke^−γre^drλ2−crλ >_{0, (3.8)}

and define

∆c(λ) =−D Z _+∞

−_∞ h(y)e^−λydy−1

+cλ+δ(0) +β(0)− ²(₁−K)β(₀)e^−γre^drλ2−crλ

1−2Ke^−γre^drλ2−crλ . (3.9) We have, forλ∈(0,λ⁺(c)),

1−2Ke^−γre^drλ2−crλ >0 and lim

λ→λ⁺(c)∆c(λ) =−_∞.

It is not difficult to see that

∆c(₀) =δ(₀) +β(₀)−²(1−K)β(0)e^−γr 1−2Ke^−γr <_0.

Furthermore, the second derivative of the functionλ7→_∆c(λ)satisfies, for allλ∈[0,λ⁺(c)),

∂²

∂λ²∆c(λ)<0.

Moreover,

d

dc[_∆c(λ)]>0 and lim

c→+_∞∆c(λ) = +_∞.

We conclude that there exists a unique c^? >0 such that

∆c^?(λ^?(c^?)) =0 and ∂

∂λ∆c^?(λ)

λ=λ^?(c^?) =0.

According to the above arguments, we have the following result.

Lemma 3.4. Assume that(2.4) and(2.5)hold. Then, there exists a unique c^? > 0and for each c >0 there exists a uniqueλ^?(c)such that

1. if c=c^?,∆c^?(λ^?(c^?)) = ^∂

∂λ∆c^?(λ)

λ=λ^?(c^?)=0,

2. if c > c^?, there exist two real roots, λ₁(c) and λ2(c), of the equation ∆c(λ) = 0 such that 0<λ₁(c)<λ₂(c)<λ⁺(c)and∆c(λ)>0for allλ∈(λ₁(c),λ₂(c)),

3. if0<c<c^?,∆c(λ)<0for allλ∈(0,λ⁺(c)).

Next, we fixc>c^? and we putλ₁ :=λ₁(c)_,λ₂ := λ₂(c)_{. We put}







κ₁(λ) =1−2Ke^−γre^drλ2−crλ, κ₂(λ) =2(1−K)β(0)e^−γre^drλ2−crλ. We need the following lemma.

Lemma 3.5. For z∈_Randλ∈ (0,λ⁺(c)), we have the following equality T(r)ξ∗e^λ·

(z−cr) = ^κ2(λ)

2(1−K)e^−γrg⁰(0)κ₁(λ)^e

λz.

Proof. We start by computing the following quantity

ξ∗e^λ·

(z−cr) =

Z _+∞

−_∞ ξ(z−cr−y)e^λydy,

=e^−λcr

+_∞ k

∑

(2Ke^−γr)^k

Z +_∞

−_∞ Γ(kr,z−y−kcr)e^λydy,

=e^−λcre^λz

+_∞ k

∑

(2Ke^−γr)^ke^{drkλ2−crkλ},

= ^e

−λcr

1−2Ke^−γre^drλ2−crλe^λz. Then,

T(r)ξ∗e^λ·

(z−cr) = ^e

−λcr

1−2Ke^−γre^drλ2−crλ Z _+∞

−_∞ Γ(r,z−y)e^λydy,

= ^e

λ²dr−λcr

1−2Ke^−γre^drλ2−crλe^λz. The proof is completed.

We prove the existence of a continuous upper and lower solutions of (3.5).

Lemma 3.6. Assume that (2.1), (2.2), (2.4),(2.5),(2.7) and(2.8) hold. Let c > c^? be fixed, with c^? given in Lemma3.4, and N^?be the positive steady state. We putλ₁ :=λ₁(c),λ₂:=λ₂(c), withλ₁(c) andλ₂(c)defined in Lemma3.4. Then, The functionφ: R →_R⁺defined by φ(z) = min{N^?,e^λ1z} is an upper solution of (3.5).

Proof. Asλ₁>0, forz₁= _λ¹

1 ln(N^?), we have φ(z) =







N^?, z≥z₁, e^λ1z, z<z₁.

(3.10)

Suppose that z ∈ [z₁,+_∞). Then, φ(z) = N^?, φ⁰(z) = φ⁰⁰(z) = 0 and as g is an increasing function on[0,N^?], we have

T(r) ξ∗g(φ)(z−cr)≤[T(r) (ξ∗g(N^?))] (z−cr) = ^g(N^?) 1−2Ke^−γr. Then, we obtain

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] + f(φ(z))−2(1−K)e^−γr

T(r) ξ∗g(φ)(z−cr)

≥ f(N^?)−2(1−K)e^−γr

g(N^?) 1−2Ke^−γr

=0.

Suppose thatz∈(−_∞,z₁). Then,φ(z) =e^λ1z. Consequently, cφ⁰(z)−D[(h∗φ)(z)−φ(z)]≥

cλ₁−D _{Z +∞}

−_∞ h(y)e^−λ1ydy−1

e^λ1z and due to (2.8)

f(φ(z))≥ f⁰(₀)φ(z) = f⁰(₀)e^λ1z.

Furthermore, we have

T(r) ξ∗g(φ)(z−cr)≤ g⁰(0)T(r) ξ∗φ

(z−cr)

≤ g⁰(0)^hT(r)ξ∗e^λ1·i

(z−cr)

= ^g

0(0)e^{drλ21+(z−cr)λ1} 1−2Ke^−γre^{drλ21−crλ1}. Then,

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] +f(φ(z))

−2(1−K)e^−γr

T(r) ξ∗g(φ)(z−cr)≥ _∆c(λ₁)e^λ1z =0.

Lemma 3.7. Assume that the hypotheses of Lemma3.6hold. Then, the functionφ: R→_R⁺ defined by φ(z) = max{0,e^λ1z−Me^ωλ1z}, with ω ∈ (1, min{2,λ₂/λ₁})and M > 1 large enough, is a lower solution of (3.5). Moreover,φ(z)≤ φ(z), for all z∈_R.

Proof. Letν∈(ω−1, min{2,λ₂/λ₁} −1). It is clear that 0<ν<1. Recall thatg(N)≤g⁰(0)N and f(N) ≥ f⁰(0)N for [0,N^?]. Under the assumption δ and β are C¹-function, there exists α>0 such that, foru∈ [_0,N^?]_,

δ(u) +β(u)−(δ(0) +β(0))≤ αu^ν and β(0)−β(u)≤ αu^ν. (3.11) We will construct a lower solutionφof the form

φ(z) =

(e^λ1z−Me^ωλ1z, z<z₂,

0, z≥z2,

with

z₂:= ¹

(ω−1)λ₁ln 1

M

,

and M > 1 is a constant. Then, z₂ < 0. First, remark that to get φ≤ φ, it suffices to choose M >(N^?)^1−ω.

Letz∈[z₂,+_∞). Then,φ(z) =0. Thus,

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] +f(φ(z))−₂(₁−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr)

= −D(h∗φ)(z)−2(1−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr). The functionφis nonnegative onR. We conclude that, for all z∈[z₂,+_∞),

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] + f(φ(z))−2(1−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr)≤0.

Letz∈(−_∞,z₂). Then,φ(z) =e^λ1z−Me^ωλ1z. We have

cφ⁰(z)−D[(h∗φ)(z)−φ(z)]≤cλ₁e^λ1z−cMωλ₁e^ωλ1z+De^λ1z−DMe^ωλ1z.

−D Z +_∞

−_∞ h(y)[e^λ1(z−y)−Me^ωλ1(z−y)]dy.

Thanks to∆c(λ₁) =0, we obtain

cφ⁰(z)−D[(h∗φ)(z)−φ(z)]≤ ^κ2(λ₁) κ₁(λ₁)^e

λ₁z− f⁰(0)e^λ1z−_∆c(ωλ₁)Me^ωλ1z

−^κ2(ωλ₁) κ₁(ωλ₁)^Me

ωλ1z+ f⁰(0)Me^ωλ1z. We know thatω∈ (1, min{2,λ₂/λ₁}), then

∆c(ωλ₁)>0 and 1−2Ke^−γre^{drω2λ21−crωλ1} >0.

By Lemma3.5, we get κ2(λ₁) κ₁(λ₁)^e

λ1z−^κ2(ωλ₁) κ₁(ωλ₁)^Me

ωλ1z ≤2(1−K)e^−γrg⁰(0)T(r)ξ∗φ

(z−cr). We conclude that

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] +f(φ(z))

≤ −_∆c(ωλ₁)Me^ωλ1z+αφ^(ν+1)(z) +2(1−K)e^−γrg⁰(0)T(r)ξ∗φ

(z−cr). It is not difficult to see that

φ^(ν+1)(_s)≤e^(ν+1)λ1s, for alls ∈_R.

Then,

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] + f(φ(z))−2(1−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr)

≤ −_∆c(ωλ₁)Me^ωλ1z+αe^(ν+1)λ1z+₂(₁−K)e^−γrαT(r)ξ∗e^(ν+1)λ1·

(z−cr)_. Using lemma3.5, we have

2(1−K)e^−γrαT(r)(ξ∗e^(ν+1)λ1·)(z−cr) = ^ακ2((ν+₁)λ₁) g⁰(0)κ₁((ν+1)λ₁)^e

(ν+1)λ1z. So,

cφ⁰(z)−D[(h∗φ)(z)−φ(z)] +f(φ(z))−2(1−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr)

≤e^ωλ1z

−_∆c(ωλ₁)M+αe^{(ν+1−ω)λ1z}

1+ ^κ2((ν+1)λ₁) g⁰(0)κ₁((ν+1)λ₁)

.

Recall thatν+1−ω >0, which implies thate^{(ν+1−ω)λ1z} <1, for allz <z₂. Then, forz<z₂, cφ⁰(z)−D[(h∗φ)(z)−φ(z)] +f(φ(z))−2(1−K)e^−γrh

T(r)ξ∗g(φ)ⁱ(z−cr)

≤e^ωλ1z

−_∆c(ωλ₁)M+α

1+ ^κ2((ν+1)λ₁) g⁰(0)κ₁((ν+1)λ₁)

. Finally, we can choose

M >maxn

1,(N^?)^1−ω,Ce[_∆c(ωλ₁)]^−1o, with

Ce:=α

1+ ^κ2((ν+1)λ₁) g⁰(0)κ₁((ν+1)λ₁)

. The proof is completed.

We define the profile set of traveling wave fronts as

Θ=







φ∈C_[0,N^?_](_R,R):

(i) φ(z) is nondecreasing on R, (ii) φ(z)≤φ(z)≤φ(z), for allz ∈_R.







Next, we state two lemmas that ensure the existence of a fixed points of operator F. For the setΘ, it is easy to see that the following lemma holds.

Lemma 3.8. The setΘis nonempty, bounded, closed and convex subset of B_ν(_R,R)with respect to the norm| · |_ν.

The proof of the following result is similar to the proof of the corresponding results in [21,27].

Lemma 3.9. Assume (2.1), (2.2), (2.4), (2.5) and(2.7) hold. Then, F(Θ) ⊂ Θ and F : Θ → Θ is continuous and compact with respect to the norm| · |_ν.

In conclusion, we get the following theorem that state the existence result.

Theorem 3.10. Assume that(2.1), (2.2), (2.4), (2.5), (2.7) and (2.8) hold. Then, for every c > c^?, (1.4)has a traveling wave front which connects(0, 0)to the positive equilibrium(N^?,u^?).

Proof. From Lemmas3.6,3.7,3.8and3.9, forc>c^?, we obtain the existence of a fixed point φ of Fbelonging toΘ, that is,

φ(z) = ¹ c

Z _z

−_∞e^{−µc(z−s)}H(φ)(s)ds. (3.12) On the other band, we have that φ(z) ≤ φ(z) ≤ φ(z) _{for all} z ∈ R, which implies that limz→−_∞φ(z) =0. Moreover,φ(z)is a nondecreasing function bounded above by N^?. Then, there exists N₀ such that lim_z→+_∞φ(z) = N₀ ≤ N^?. Recall that 0 ≤ φ(z) 6≡ 0 for all z ∈ _R.

This implies that N₀ ∈(_0,N^?]. Using L’Hospital’s rule for (3.12), we obtain N₀ = lim

z→+_∞φ(z) = lim

z→+_∞

1

µH(φ)(z),

= ¹ µ

µN₀− f(N₀) +²(1−K)e^−γr 1−2Ke^−γr g(N₀)

.

We deduce that(1−2Ke^−γr)f(N₀) =2(1−K)e^−γrg(N₀). Since we have the uniqueness of the positive steady state, we conclude that N₀ = N^?. As a consequence, we get the existence of traveling wave front satisfying (3.6). The proof is complete.

The following theorem concerns the result for the critical velocity.

Theorem 3.11. Assume that(2.1),(2.2),(2.4),(2.5),(2.7)and(2.8)hold and c=c^∗. Then, Equation (1.4)has a traveling wave front which connects(0, 0)to the positive equilibrium(N^?,u^?).

Proof. Let c = c^?. We use some ideas developed in [24,34,35]. We consider a sequence (c_m)_m≥1 ⊆ (c^?,+_∞)such that lim_m→+_∞c_m = c^?. For instance, we can choosec_m =c^?+1/m.

It follows from Theorem3.10that for c=c_m > c^?, (1.4) has a solution inΘ. We denote byφ_m

this solution. Without loss of generality, we may assume that φ_m(0) = N^?/2. Furthermore, φ_m is given by

φ_m(z) = ¹ c_m

Z _z

−_∞e^{−cmµ(z−s)}H(φ_m)(s)ds.

We can verify the boundedness ofφ⁰_m onR by differentiating the above equality with respect toz. It follows thatφm is uniformly bounded and equicontinuous sequences of functions on R. By Ascoli’s theorem there exists a subsequence of (c_m)_m≥1 (for simplicity, we preserve the same sequence (c_m)_m≥1), such that limm→+_∞c_m = c^? and φ_m(z) converge uniformly on every bounded interval. Then, they converge pointwise on R to φ(z). By using Lebesgue’s dominated convergence theorem, we get

φ(z) = ¹ c^?

Z _z

−_∞e^{−cµ?(z−s)}H(φ)(s)ds.

Then,φ is a solution of (3.5) with c= c^?. It is not difficult to see that φis nondecreasing on R and satisfying φ(0) = N^?/2 and 0 ≤ φ(z) ≤ N^? for all z ∈ _R. Then, lim_z→−_∞φ(z) and limz→+_∞φ(z)exist. Obviously, limz→−_∞φ(z) =0 and limz→+_∞φ(z) = N^?. As a consequence, we have that forc=c^?, (3.5) has a solution inΘ. The proof is complete.

In the next results, we give some properties of the traveling wave fronts of (3.5).

Lemma 3.12. Letφbe a traveling wave front of (3.5)connecting0to the positive equilibrium N^?and let z∈_R. Then, we have

1.

Z _z

−_∞|(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)|dy<+_∞, 2. ϕ(z):=

Z _z

−_∞φ(y)dy<+_∞, 3.

Z _z

−_∞[T(r) (ξ∗φ)] (y−cr)dy= ((_Γ∗ξ)∗ϕ) (z−cr), 4.

Z _z

−_∞|(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)|dy< +_∞.

Proof. (1) The definition of convolution product implies, fort< z, Z _z

t

(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy

=

Z _z

t

(1−2Ke^−γr)

Z _+∞

−_∞

(_Γ∗ξ) (l)φ(y−cr−l)dl−φ(y)dy.

We can check that(1−2Ke^−γr)R+_∞

−_∞ (_Γ∗ξ) (l)dl =1. Then, we can write the following equal- ity

Z _z

t

(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy

= (1−2Ke^−γr)

Z _z

t

Z +_∞

−_∞ (_Γ∗ξ) (l) [φ(y−cr−l)−φ(y)]dldy.

Moreover, we have

φ(_y−cr−l)−φ(_y) =−

Z _y

y−cr−l

φ⁰(_x)_dx= −(l+_cr)

Z ₁

0

φ⁰(_y−η(_l+_cr))_dη.

Then, we get Z _z

t

(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy

−(1−2Ke^−γr)

Z _z

t

Z +_∞

−_∞ (l+cr) _{Z 1}

0 φ⁰(y−η(l+cr))dη

(_Γ∗ξ) (l)dldy.

Fubini’s theorem with the dominated convergence theorem implies Z _z

−_∞

(₁−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy

=−(₁−2Ke^−γr)

Z _+∞

−_∞

(l+cr)

Z ₁

0

t→−lim_∞ Z _z

t

φ⁰(y−η(l+cr))dy

dη(_Γ∗ξ) (l)dl.

By using the fact that lim_x→−_∞φ(x) =0, we get

t→−lim_∞ Z _z

t φ⁰(y−η(l+cr))dy=φ(z−η(l+cr)). This yields

Z _z

−_∞(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy

=−(1−2Ke^−γr)

Z +_∞

−_∞ (l+cr)

Z ₁

0 φ(z−η(l+cr))dη(_Γ∗ξ) (l)dl.

The function (l,z) ∈ _R×_R 7−→ R1

0 φ(z−η(l+cr))dη is bounded. We have also that R+_∞

−_∞ |l (_Γ∗ξ) (l)|dl <+∞. Then, forz∈ _R,

Z _z

−_∞|(1−2Ke^−γr) [_T(_r) (ξ∗φ)] (y−cr)−φ(_y)|dy< +_∞.

(2) The functionφis positive and satisfies

cφ⁰(z) =D[(h∗φ)(z)−φ(z)]− f(φ(z)) +2(1−K)e^−γr[T(r) (ξ∗g(φ))] (z−cr), (3.13) with φ(−_∞) =0, φ(+_∞) =N^?. The continuity ofβandδimplies that

y→−lim_∞β(φ(y)) =β(₀) _{and lim}

y→−_∞δ(φ(y)) +β(φ(y)) =δ(₀) +β(₀)_.

Then, forε>0 small enough (ε< β(0)), there existsy_ε <0 such that, for ally<y_ε, we have (

β(0)−ε< β(φ(y))≤β(0) +ε,

δ(0) +β(0)−ε ≤δ(φ(y)) +β(φ(y))< δ(0) +β(0) +ε.

Then, (3.13) implies, fory <y_ε,

cφ⁰(y)≥D[(h∗φ)(z)−φ(z)]−(ε+δ(0) +β(0))φ(y)

+2(1−K)e^−γr(β(0)−ε) [T(r) (ξ∗φ)] (z−cr). (3.14) Let us denote by

J_ε = ²(1−K)e^−γr

1−2Ke^−γr (β(0)−ε)−(ε+δ(0) +β(0)), with 0<ε< β(0).

We rewrite the inequality (3.14), for y<y_ε, in the form Jεφ(y)≤cφ⁰(y)−D[(h∗φ)(z)−φ(z)]

− ²(1−K)e^−γr

1−2Ke^−γr (β(0)−ε)[(1−2Ke^−γr) [T(r) (ξ∗φ)] (z−cr)−φ(y)]. On the other hand, the condition for the existence of positive equilibrium can be written as

2(1−K)e^−γr

1−2Ke^−γr β(0)−(δ(0) +β(0))>0.

Then, we can chooseε∈(0,β(0))small enough such that ε

2(1−K)e^−γr 1−2Ke^−γr +1

< ²(1−K)e^−γr

1−2Ke^−γr β(0)−(δ(0) +β(0)). Hence, Jε is positive. As a consequence, for z<y_ε,

0≤ Jε

Z _z

−_∞φ(y)dy≤cφ(z)−D Z _z

−_∞[(h∗φ)(y)−φ(y)]dy

−²(1−K)e^−γr

1−2Ke^−γr (β(0)−ε)

Z _z

−_∞(1−2Ke^−γr) [T(r) (ξ∗φ)] (y−cr)−φ(y)dy< +_∞.

(3.15)

Then, for allz∈_{R, 0}≤

Z _z

−_∞φ(y)dy <+_∞.

(3) By Fubini’s theorem, we can check that, fort<z, Z _z

t

[T(r) (ξ∗φ)] (y−cr)dy=

Z +_∞

−_∞ (_Γ∗ξ) (z−cr−y)

Z _y

t φ(s)dsdy.

By using the dominated convergence theorem, we obtain Z _z

−_∞[T(r) (ξ∗φ)] (y−cr)dy=

Z _+∞

−_∞ (_Γ∗ξ) (x)

t→−lim_∞ Z _z

t φ(y−cr−x)dy

dx,

=

Z +_∞

−_∞ (_Γ∗ξ) (x)

Z _z−cr−x

−_∞ φ(s)dsdx,

= ((_Γ∗ξ)∗ϕ) (z−cr). (4) By the same techniques as in the proof of (1), we have

Z _z

−_∞

(₁−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy

= (1−2Ke^−γr)

Z _z

−_∞ Z _+∞

−_∞ (_Γ∗ξ) (l) [ϕ(y−cr−l)−ϕ(y)]dldy.

Then, we get Z _z

−_∞(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy

=−(1−2Ke^−γr)

Z _z

−_∞ Z +_∞

−_∞ (l+cr) _{Z 1}

0 ϕ⁰(y−η(l+cr))dη

(_Γ∗ξ) (l)dldy.

Then, we obtain Z _z

−_∞(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy

=−(1−2Ke^−γr)

Z +_∞

−_∞ (l+cr) _{Z 1}

0 ϕ(z−η(l+cr))dη

(_Γ∗ξ) (l)dl.

(3.16)

We have proved Z _z

−_∞|(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)|dy< +_∞.

Now, let us considerφsolution of (3.5) satisfying (3.6). The next proposition establish the asymptotic behavior of the profile φ(z)whenz → −_∞.

Proposition 3.13. There exists a positive constantµ₀ <λ⁺(c)such thatφ(z) =O(e^µ0z)as z→ −_∞.

Moreover,

sup

z∈_R

e^−µ0zφ(z)<+_{∞. (3.17)} Proof. From the Lemma3.12and the inequality (3.15), we have, fory<y_ε

0≤ J_εϕ(y)≤cφ(y)−D Z _y

−∞[(h∗φ)(s)−φ(s)]ds

−²(1−K)e^−γr

1−2Ke^−γr (β(0)−ε)(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)<+_∞.

(3.18)

By integrating the both sides of (3.18), from −_∞to z≤y_ε, we obtain Jε

Z _z

−_∞ϕ(y)dy≤cϕ(z)−D Z _z

−_∞ Z _y

−_∞[(h∗φ)(s)−φ(s)]dsdy

− ²(1−K)e^−γr

1−2Ke^−γr (β(0)−ε)

Z _z

−_∞(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy.

Thanks to (3.16), we have

−

Z _z

−_∞(1−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy

= (1−2Ke^−γr)

Z +_∞

−_∞ (l+cr) _{Z 1}

0 ϕ(z−η(l+cr))dη

(_Γ∗ξ) (l)dl.

The functionη∈[0, 1]7→ (s+cr)ϕ(z−η(s+cr))is decreasing. Then, we obtain

−

Z _z

−_∞

(₁−2Ke^−γr) ((_Γ∗ξ)∗ϕ) (y−cr)−ϕ(y)dy

≤ (1−2Ke^−γr)ϕ(z)

Z _+∞

−_∞

(l+cr) (_Γ∗ξ) (l)dl.

Moreover, from the fact that φ⁰(z)≥0 forz∈ _{R, we have}

Z _z

−_∞ Z _y

−_∞

[(h∗φ)(s)−φ(s)]dsdy=

Z _z

−_∞

(h∗ϕ)(y)−ϕ(y)]dy,

=

Z _z

−_∞ Z _+∞

−_∞ h(t)[ϕ(y−t)−ϕ(y)]dtdy,

=

Z _z

−_∞ Z _+∞

0 h(t)[ϕ(y+t) +ϕ(y−t)−2ϕ(y)]dtdy,

=

Z _z

−_∞ Z _+∞

0 h(t)

Z _y

y−t

[φ(t+s)−φ(s)]dsdtdy,

≥0.